Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 + 21\cdot 257 + 142\cdot 257^{2} + 138\cdot 257^{3} + 153\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 90 + 102\cdot 257 + 212\cdot 257^{2} + 217\cdot 257^{3} + 210\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 107 + 6\cdot 257 + 204\cdot 257^{2} + 210\cdot 257^{3} + 181\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 117 + 113\cdot 257 + 152\cdot 257^{2} + 253\cdot 257^{3} + 223\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 148 + 13\cdot 257 + 60\cdot 257^{2} + 207\cdot 257^{3} +O\left(257^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
